water

Article

A Two-Dimensional Depth-Averaged SedimentTransport Mobile-Bed Model with Polygonal Meshes

Yong G. Lai

Technical Service Center, U.S. Bureau of Reclamation, Bldg. 67, P.O. Box 25007, Denver, CO 80225, USA;ylai@usbr.gov; Tel.: +1-303-445-2560

Received: 6 March 2020; Accepted: 1 April 2020; Published: 4 April 2020�����������������

Abstract: A polygonal-mesh based numerical method is developed to simulate sediment transport inmobile-bed streams with free surfaces. The flow and sediment transport governing equations aredepth-averaged and solved in the two-dimensional (2D) horizontal space. The flow and sedimenttransport are further coupled to the stream bed changes so that erosion and deposition processesare simulated together with the mobile bed changes. Multiple subsurface bed layers are allowedso that bed stratigraphy may be taken into consideration. The proposed numerical discretizationis valid for the most flexible polygonal mesh type which includes all existing meshes in use suchas the quadrilateral-triangle hybrid mesh. The finite-volume method is adopted such that the massconservations of both water and sediment are satisfied locally and globally. The sediment transportand stream bed processes are formulated in a general way so that the proposed numerical methodmay be applied to a wide range of streams and suitable for practical stream applications. The technicaldetails of the numerical method are presented; model verification and validation studies are reportedusing selected cases having physical model or field measured data. The developed model is intendedfor general-purpose use available to the public.

Keywords: sediment transport model; mobile-bed model; scour and erosion; 2D depth-averagedmodel; polygonal mesh

1. Introduction

Sediment transport in streams and sedimentation in reservoirs and the associated morphologicalchanges are fundamentally important for the success of many environmental projects. Relevant areasinclude stream restoration for environmental and ecological benefits, reservoir sedimentation andits sustainability, flood risk management, dam removal, estuary and coastal management, and localscours around, e.g., bridge piers, among many others [1].

One-dimensional (1D) hydraulic flow models have been widely used in practical hydraulicprojects for half a century; 1D mobile-bed models have also been developed since and applied inpractical projects in the past two decades. Some of the 1D flow and mobile-bed models in wide useinclude HEC-RAS [2,3], MIKE11 [4], CCHE1D [5], and SRH-1D [6]. 1D mobile-bed models will remainuseful for a foreseeable future, particularly for applications with a large spatial extent and/or overa long time period. Their limitations, however, are well known. In recent years, many engineeringapplications are turning to multi-dimensional models. Among the category, the two-dimensional(2D) depth-averaged models are gaining widespread acceptance due to recent numerical algorithmdevelopments and computer hardware advancements [7,8].

2D depth-averaged flow models have been developed since the works of Chow and Ben-Zvi [9]and Kuipers and Vreugdenhil [10]. 2D mobile-bed models were developed since. Earlier works includeCelik and Rodi [11], Spasojevic and Holly [12], Minh Duc [13], Olsen [14], and Wu et al. [15], amongothers. These early models adopted simple meshes and often assumed local equilibrium in sediment

Water 2020, 12, 1032; doi:10.3390/w12041032 www.mdpi.com/journal/water

Water 2020, 12, 1032 2 of 21

load transport. More general 2D mobile-bed models have been developed since. For example, Wu [16]developed an unsteady model allowing non-uniform and non-equilibrium sediment transport forboth suspended load and bedload. The model was tested against several experimental and fieldcases; good agreements were reported. The model, however, was limited to the structured curvilinearmesh. Hung et al. [17] proposed an implicit two-step operator-splitting method to solve the governingequations. The mesh type was, however, limited to the structured, orthogonal curvilinear coordinatesystem. Huang et al. [18] developed an adaptive mesh refinement model for dam-break flows overmobile-bed streams. The model was validated against experimental cases with good agreements.The model was limited to the rectangular mesh though adaptation was allowed. At present, there areonly a limited number of 2D mobile-bed models which are available to the public for general uses.Such models include CCHE2D [19], TELEMAC [20], UnTRIM [21,22], and Delft3D [23]. CCHE2D wasbased on the finite element discretization method using the purely quadrilateral or purely triangularmeshes; TELEMAC and UnTRIM adopted the purely triangular meshes; and Delft3D was based on thestructured mesh. A more flexible mesh version of Delft3D was reported by Kernkamp et al. [24] andused for the suspended sediment modeling in the San Francisco delta and estuary [25]. Orthogonalquadrilaterals, however, were highly recommended according to the developers [26]. In 2D mobile-bedmodeling, recent attentions have been on the fundamental formulation issue as discussed by Iversonand Ouyang [27], Cao et al. [28], and Liu et al. [29]. A generally applicable sediment transportformulation is yet to be agreed upon and developed.

Many existing models did not address the unsteady nature of the sediment transport: thequasi-steady assumption has often been made such that the use of the Exner equation were justified [30].However, a truly unsteady model is not only more general, it is also needed for many processes suchas the sediment load travel after a sudden release of sediments due to dam removal or dam failure.Most existing sediment models classified a sediment load into suspended or bed load; but in reality,the transport mode is dependent on local flow conditions and may switch between the two on anatural stream. Further, the switch is not sudden and there usually exist mixed modes in-between thetwo. Existing sediment models often assumed that the sediment transport rate equaled the sedimenttransport capacity—the so-called equilibrium assumption. This assumption is appropriate mainly forlarge spatial and time scales such as 1D modeling; it may fail for event-based sediment transport aswell as cases of significant erosion [16]. Some simplifications have been discussed by Cao et al. [28]who pointed out the need for true unsteady and non-equilibrium modeling if a model is intended forgeneral uses. A most significant restriction of the existing models is the adoption of the mesh typewhich is too restrictive, as reviewed above.

In this study, a 2D depth-averaged mobile-bed model is developed with the aim of the model to becomprehensive and general-purpose. It is being available freely to the public. A major contribution ofthe present model is the adoption of a general mesh type—the polygon-based mesh—for the sedimenttransport and mobile-bed modeling. Other new contributions are related to the treatment of the majorphysical processes: as-general-as-possible approaches are adopted. They include: single sedimenttransport equation (total load) for all modes (suspended, mixed or bed load); truly unsteady andtightly-coupled formulation of all governing equations, multi-size sediment partition (an arbitrarynumber of sediment size classes may be used), and non-equilibrium sediment transport. Details ofthese physical processes are explained below in the governing equations section. The proposed modelis applicable to a wide range of environmental stream and reservoir sediment transport modelingissues. To our knowledge, no 2D depth-averaged mobile-bed models have been published that adoptsthe polygonal meshes.

2. Governing Equations

A 2D depth-averaged mobile-bed model, in general, consists of four interrelated process modules:hydraulic flow, sediment transport, mobile-bed dynamics, and bank erosion. Only the first three

Water 2020, 12, 1032 3 of 21

modules are described as no bank erosion process is considered in this study. The bank erosionmodeling was described by Lai [31] if readers are interested.

2.1. Flow Equations

The flow solver is based on the verified model of Lai [32]. Details may be found from thatreference; only the governing equations are presented herein. The 2D depth-averaged flow equationsare as follows:

∂h∂t

+∂(hU)

∂x+∂(hV)

∂y= 0 (1)

∂hU∂t

+∂hUU∂x

+∂hVU∂y

= −gh∂z∂x

+∂(hTxx)

∂x+∂(hTxy

)∂y

−τbxρ

(2)

∂hV∂t

+∂hUV∂x

+∂hVV∂y

= −gh∂z∂y

+∂(hTxy

)∂x

+∂(hTyy

)∂y

−τby

ρ(3)

In the above, x and y are horizontal Cartesian coordinates, t is time, h is water depth, U and V aredepth-averaged velocity components in x and y directions, respectively, g is gravitational acceleration,Txx, Txy and Tyy are depth-averaged stresses due to turbulence and dispersion, z = zb + h is watersurface elevation, zb is bed elevation, ρ is water density, and τbx and τby are bed shear stresses. The bedstresses are computed by the Manning’s equation:

(τbx, τby

)= ρU2

∗

(U, V)√

U2 + V2= ρC f

√U2 + V2(U, V) (4)

where C f = gn2/h1/3, n is Manning’s coefficient, and U∗ is bed frictional velocity. Effective stresses arecomputed by:

Txx = 2(ν+ νt)∂U∂x

(5a)

Tyy = 2(ν+ νt)∂V∂y

(5b)

Txy = (ν+ νt)

(∂U∂y

+∂V∂x

)(5c)

where ν is kinematic viscosity of water, and νt is eddy viscosity of turbulence.The turbulence eddy viscosity needs a turbulence model. Two models are adopted [33]: the

depth-averaged parabolic model and the two-equation k-εmodel. With the parabolic model, the eddyviscosity is calculated by νt = CtU∗h and the frictional velocity U∗ is defined in Equation (4). The modelconstant Ct may range from 0.3 to 1.0; but the default value of Ct= 0.7 is used in this study. The k-modelcomputes the eddy viscosity by νt = Cµk2/ε and the two additional partial difference equations fork and ε are solved. The two turbulence equations are not presented herein as the validation casessimulated in this study use the parabolic model only and readers may refer to Lai [32] for further details.

2.2. Sediment Transport Equations

The water column and stream bed are divided into three vertical zones: water column, activelayer, and subsurface. Sediments in the water column are transported by flowing water and accordingto the mass conservation principle. The active layer is a special zone consisting of a thin sediment layerbetween the water column and the underneath subsurface. Within the active layer, sediment exchangetakes place between sediments in the water column and those in the subsurface. The subsurfaceincludes the bed materials underneath the active layer; it may be divided further into multiplesubsurface layers so that the vertical variation of sediment composition (stratigraphy) may be takeninto account. Physical processes within each zone are different and modeled separately.

Water 2020, 12, 1032 4 of 21

Sediments in the three zones may be divided into a user-specified number of size classes, sayNsed. In the water column, each size class k is governed by the following single equation no matter thetransport mode (suspended, mixed or bed load):

∂hCk∂t

+∂ cos(αk)βkVthCk

∂x+∂sin(αk)βkVthCk

∂y=

∂∂x

(h fkDx

∂Ck∂x

)+

∂∂y

(h fkDy

∂Ck∂y

)+ Sek (6)

In the above, subscript k denotes that the variable is for sediment size class k, Ck is thedepth-averaged sediment concentration by volume, βk = Vsed,k/Vt is the sediment-to-flow velocityratio, Vt =

√U2 + V2 is the depth-averaged flow velocity, αk is the angle of the sediment transport

direction relative to x-axis, fk is the transport mode parameter representing the suspended load fraction,Dx and Dy are the sediment mixing coefficients in the x- and y-directions, respectively, and Sek is thesediment exchange rate between sediments in the water column and those in the active layer or on thebed. The above equation was derived from the mass conservation law by Greimann et al. [30] and tookthe non-equilibrium nature of a sediment load into account.

The sediment transport angle (αk) may deviate from flow velocity direction due to secondaryflows and gravity forces on a transverse bed slope. Several approaches may be used to take theseeffects into account [34]. In this study, the approach of Struiksma and Crosato [35] is adopted. That is,the angle is computed by:

tan(αk) =

sin(δk) −(1− fk)Cg1

0.85√θk

∂Zb∂y

cos(δk) −(1− fk)Cg1

0.85√θk

∂Zb∂x

(7)

In the above, θk = τb/[ρg(s− 1)dk] is the Shields parameter (τb is the bed shear stress, s = ρsρ − 1,

ρs is the sediment density, dk is the sediment diameter for size class k), Cg1 is the particle shape factor,and δk is the angle of the bed shear stress. The study of Talmon et al. [36] suggested that Cg1 rangedfrom 0.5 to 1.0. The bed shear stress angle includes the flow direction and secondary flow effect and iscomputed by:

δk = tan−1(V

U

)− (1− fk)tan−1

[2Cspi

κ2

(1−

n√

g

κh1/6

)h

Rc

](8)

In the above, κ is the von Karman constant (0.41), Rc is the local radius of curvature of flowstreamlines, and Cspi is a model coefficient (1.0).

The sediment transport mode parameter fk is introduced to represent the percentage of sedimentstransported as the suspended load. A similar parameter was introduced by Holly and Rahuel [37] asthe “allocation coefficient.” An empirical equation developed by Greimann et al. [30] is used as:

fk = Min(1.0, 2.5e−Zk

)(9)

In the above, Zk = ωsed,k / (κU∗) is the suspension parameter and ωsed,k is the particle fall velocity.If bed load is dominant, fk = 0 is used; fk = 1 is used for suspended load.

The ratio of sediment-to-flow velocity, βk, was assumed to be 1.0 by most previous studies, which isadequate for many applications. For some applications, such as the unsteady movement of a specifiedsediment load from a reservoir outlet or a plug, the ratio is not 1.0, and an empirical relation should bedeveloped. In this study, the modified equation of Greimann et al. [30] is used as:

βk = Max(βk,sus, βk,bed

)(10a)

βk,bed =U∗Vt

1.1Φ0.17k [1− exp(−5Φk)]

√θr

; Φk =θkθr< 20 (10b)

βk,sus = 1 +U∗

2κVt[1− exp(2.7Zk)]; Zk < 1.0 (10c)

Water 2020, 12, 1032 5 of 21

where θr is the reference Shields parameter (0.045).The sediment rate term is related to the sediment transport capacity as follows:

Se,k =1

Lt,k

(q∗t,k −VthCk

)(11a)

Lt,k = (1− fk)Lb,k + fkζVth/ωs,k (11b)

In the above, q∗t,k is the sediment transport capacity (volume sediment rate per unit width), Lb,k isthe bed load adaptation length, and ς is the suspended load parameter.

The sediment transport capacity may be computed with an extensive number of existing sedimentcapacity or equilibrium equations. Many have been implemented in the present model; but only twoare used in the present model verification studies: Engelund and Hansen [38] equation for sandystreams and Parker [39] equation for gravel and mixed sand-gravel streams. The Engelund-Hansenequation is expressed as:

q∗t,k√sgd3

k

= 0.05pakV2

t

gdk√

s(s− 1)

[τb

(s− 1)ρgdk

]1.5

(12)

where pak is the volume fraction of sediment size class k in the active layer. The Parker equation wasoriginally developed for gravel streams but was later found to be applicable also to sand and gravelmixture [40]. The equation is expressed as:

q∗t,k g(s− 1)

(τb/ρ)1.5= pkG(Φk) (13a)

Φk =θkθc

(dkd50

)α(13b)

In the above, pk is the volumetric fraction of the kth sediment size class in the bed, θc is criticalShield’s parameter, d50 is the medium diameter of the sediment mixture on the bed, and α is theexposure factor. The function in (13a) was fit to the field data and is expressed as:

G(Φ) =

11.933 (1− 0.853/Φ)4.5, Φ > 1.590.00218 exp

[14.2(Φ − 1) − 9.28(Φ − 1)2

], 1.0 ≤ Φ ≤ 1.59

0.00218Φ14.2, Φ < 1.0(14)

Parker capacity equation allows two parameters to be defined: θc and α. θc represents a criticalreference stress above which sediment is mobilized; α is the exposure factor to account for the hidingeffect. Hiding is effective for a sediment mixture in which the critical shear stress is reduced forlarger particles and increased for smaller particles. Ideally, these two parameters should be fit toavailable data for a specific stream under simulation. Without site specific data, several references mayprovide guidance such as Komar [41], Buffington and Montgomery [42], Andrews [40], and Wilcoxand Crowe [43]. In general, θc varies from 0.03 to 0.08 and α from 0.11 to 0.67. In this study, θc = 0.045and α = 0.65 are used.

The non-equilibrium bedload adaptation length characterizes the distance for bedload sedimentsto adjust from a non-equilibrium to equilibrium state; it is related to the scales of the sediment transport,bedform, and stream geometry. It is also a function of the sediment size such that an increase in sizeleads to a decrease in the adaptation length. A number of methods have been proposed. For example,Thuc [44] applied the sand ripple length, Rahuel et al. [45] used the numerical mesh size, and Wu [16]recommended the dominant length of bedforms such as sand dunes and alternate bars. A reviewhas been conducted by Gaeuman et al. [46]. In the present study, a constant length scale (multiple

Water 2020, 12, 1032 6 of 21

times of the stream width) is used for gravel streams while the saltation length formula of Philips andSutherland [47] is used for sandy streams. The Philips-Sutherland equation is expressed as:

Lb,k = Csl(θk − θc)dk (15)

where θk > θc = 0.045 is assumed, and Csi is a model constant with a value of 4000.Determination of the suspended sediment parameter (ζ) relies on empirical data. Studies [48,49]

suggested that a constant ζ ranging from 0.25 to 1.0 might be used and its value depended on whetherthe bed experienced net deposition or erosion. The recommended value was ζ = 1.0 for net erosionand ζ = 0.25 for net deposition.

Finally, the mixing coefficients, Dx and Dy, include contributions from both turbulence as well asdispersion. For many cases, zero mixing coefficients may be used. Otherwise, the coefficients are set toequal to the turbulent viscosity with the Schmidt number specified by a user.

2.3. Mobile-Bed Equations

The elevation of a mobile-bed surface changes due to net erosion and deposition. Elevationchanges are contributed by all sediment size classes and computed by the net sediment exchangesbetween those in the water column and those in the active layer. The change in Zb due to sediment sizeclass k obeys the following equation:

ηak

(∂Zb∂t

)k= −

.Vk = −

q∗t,k −VthCk

Lt,k(16)

where ηak = 1 − σak is the porosity parameter of the active layer, σak is the porosity for the k-th sizeclass in the active layer, and

.Vk is the net volume erosion rate per unit area for size class k.

Other mobile-bed processes considered include the sediment exchanges and gradation changesin the subsurface layers. The active layer is the top bed layer participating in the sediment exchangebetween the water column and the subsurface while the subsurface layers provide sediments to orreceive sediments from the active layer. In this study, the volume fraction and porosity of the activeand subsurface layers are the dependent variables. Their governing equation are derived from themass conservation law. The volume fraction changes in the active layer is given as:

∂mapak

∂t= −

.Vk + p1k

∑i

.Vi if net erosion

∑i

.Vi ≥ 0

(17a)

∂mapak

∂t= −

.Vk + pak

∑i

.Vi if net deposition

∑i

.Vi < 0

(17b)

In the above, ma is the total sediment volume in the active layer, pak is the volume fraction of k-thclass in the active layer, p1k is the volume fraction of k-th class in the first subsurface layer (beneaththe active layer). The total volume per unit area in the active layer (ma) is kept constant throughoutthe simulation. This is in contrast to previous studies in which the mass was kept constant e.g., [16].The active layer volume is a user input via the active layer thickness (δa). In general, the active thicknessδa takes the value of Nad90 with Na ranging from 1.0 for large boulders to 20.0 for fine sands.

The porosity of the active layer is governed by the volume conservation equation—a kinematicconstraint—and expressed as:

∂δak∂t

= −

.Vk

η̃k+ p1k

∑i

.Vi

η2kif

∑i

.Vi ≥ 0 (18a)

Water 2020, 12, 1032 7 of 21

∂δak∂t

= −

.Vk

η̃k+ pak

∑i

.Vi

ηakif

∑i

.Vi < 0 (18b)

In the above, η̃k is computed by:η̃k = ηak if

.Vi ≥ 0 (19a)

η̃k = ηsk if.

Vi < 0 (19b)

and ηsk is the porosity parameter for the suspended sediments in the water column.The volume fraction (pLk), the porosity parameter (ηLk), and the thickness (tL) of subsurface layer

L (1 to the total number of subsurface layers) are continuously updated during a simulation. In thisstudy, the first subsurface layer immediately underneath the active layer exchanges sediments withthe active layer so that the total volume of the active layer is maintained. As a result, the thicknessof the immediate subsurface layer may increase or decrease. The remaining subsurface layers areunaltered until the upper subsurface layer is depleted completely. When it occurs, the lower subsurfacelayer plays the role of the upper layer unless all specified subsurface layers are eroded. For thefirst subsurface layer, termed layer 1, the volume fraction (p1k), the porosity parameter (η1k), and itsthickness need to be re-computed. For net erosion, p1k and η1k do not change but the thickness changeis governed by:

dt1kdt

= −

∑i

.Vi

∑

i

p1i

η1i

(20)

where subscript i runs through all sediment size classes. For net deposition, the thickness change isgoverned by:

dt1kdt

= −

∑i

.Vi

∑

i

pai

ηai

(21)

And p1k and η1k are updated by fully mixing the new depositions from the active layer with thesediments in layer 1.

3. Numerical Methods

The sediment module and flow solver are linked through the tightly coupled approach. The sametime step is used for both the flow and sediment processes. Within a time step, multiple iterations areexecuted. At a new iteration, the flow equations are solved first assuming stream bed is at the previousiteration; the sediment transport and mobile-bed dynamic equations are solved next using the flowfield computed at the new iteration.

All governing equations are discretized using the finite-volume method, following the works ofLai et al. [50] and Lai [32]. The solution domain is covered with an unstructured mesh with eachmesh cell assuming a polygonal shape. All dependent variables are stored at the geometric center of apolygon. The governing equations are integrated over a polygon using the Gaussian theorem to obtainthe discretized equation set. The numerical method of the flow solver has been described by Lai [32] andis omitted herein. Only the numerical method of the sediment transport Equation (6) is discussed below.

The sediment transport Equation (6) may be generally expressed as follows:

∂hΦ∂t

+∇·(h→

VΦ)= ∇·(Γ∇Φ) + S∗Φ (22)

Here Φ denotes a sediment dependent variable, Γ is the diffusivity, and S∗Φ is the source/sink term.Integration over an arbitrarily shaped polygon P shown in Figure 1. leads to:(

hn+1P Φn+1

P − hnPΦn

P

)A

∆t+

∑all−sides

(hCVC|

→s |)n+1

Φn+1C =

∑all−sides

(Γn+1

C ∇Φn+1·→n |→s |)+ SΦ (23)

Water 2020, 12, 1032 8 of 21

In the above, ∆t is time step, A is polygon area, VC =→

VC·→n is the velocity component normal

to the polygon side (e.g., P1P2 in Figure 1) and evaluated at the side center C,→n is polygon side unit

normal vector,→s is the polygon side distance vector (e.g., from P1 to P2 in Figure 1), and SΦ = S∗ΦA.

8 of 19

C, �⃗� is polygon side unit normal vector, 𝑠 is the polygon side distance vector (e.g., from P1 to P2 in 324 Error! Reference source not found.), and 𝑆Φ = 𝑆Φ

∗ 𝐴. 325

326

Figure 1. Schematic illustrating a polygon P along with one of its neighboring polygons N. 327

Subscript C indicates a value evaluated at the center of a polygon side and superscript, n or n+1, 328 denotes the time level. In the remaining discussion, superscript n+1 will be dropped for ease of 329 notation. Note that the first-order Euler implicit time discretization is adopted. The main task of the 330 discretization is to obtain appropriate expressions for the convective and diffusive fluxes at each 331 polygon side. 332

Discretization of the diffusion term, the first on the right-hand side of Equation (23), needs 333

further attention. The final expression for can be written as: 334 335 ∇Φ ∙ �⃗� |𝑠 | = 𝐷𝑛(Φ𝑁 − Φ𝑃) + 𝐷𝑐(Φ𝑃2 − Φ𝑝1) (24a) 336

337

𝐷𝑛 =|𝑠 |

(𝑟 1+𝑟 2)∙�⃗� ; 𝐷𝑐 = −

(𝑟 1+𝑟 2)∙𝑠 /|𝑠 |

(𝑟 1+𝑟 2)∙�⃗� (24b) 338

339 In the above, 𝑟 1 is the distance vector from P to C and 𝑟 2 is from C to N. The normal and cross 340

diffusion coefficients, 𝐷𝑛 and 𝐷𝑐, at each polygon side involve only geometric variables; they are 341 calculated only once in the beginning of the computation. 342

Calculation of a variable, say Y, at the center C of a polygon side is discussed next. This is an 343 interpolation operation used frequently for variables. A second-order accurate expression is derived 344 below. As shown in Error! Reference source not found., a point I is defined as the intercept point 345 between line PN and line P1P2. A second-order interpolation for point I may be derived to be: 346

347

𝑌𝐼 = 𝛿1𝑌𝑁+𝛿2𝑌𝑃

𝛿1+𝛿2 (25) 348

349 with 𝛿1 = 𝑟 1 ∙ 𝑛⃗⃗⃗⃗ and 𝛿2 = 𝑟 2 ∙ 𝑛⃗⃗⃗⃗ . 𝑌𝐼 may be used to approximate the value at the side center C. 350

This treatment, however, does not guarantee second-order accuracy unless 𝑟 1 and 𝑟 2 are parallel. 351 A truly second-order expression is derived to be: 352

353 𝑌𝐶 = 𝑌𝐼 − 𝐶𝑠𝑖𝑑𝑒(𝑌𝑃2 − 𝑌𝑃1) (26a) 354 355

𝐶𝑠𝑖𝑑𝑒 = (𝛿1𝑟 2−𝛿2𝑟 1)∙𝑠

(𝛿1+𝛿2)|𝑠 |2 (26b) 356

357 Φ𝑐 in the convective term of Equation (23) adopts the second-order scheme with a damping 358

term. It is derived by blending the first-order upwind scheme with the second-order central 359 difference scheme and may be expressed as: 360

361

n•

Figure 1. Schematic illustrating a polygon P along with one of its neighboring polygons N.

Subscript C indicates a value evaluated at the center of a polygon side and superscript, n or n+ 1, denotes the time level. In the remaining discussion, superscript n + 1 will be dropped for easeof notation. Note that the first-order Euler implicit time discretization is adopted. The main task ofthe discretization is to obtain appropriate expressions for the convective and diffusive fluxes at eachpolygon side.

Discretization of the diffusion term, the first on the right-hand side of Equation (23), needs furtherattention. The final expression for ∇Φ·

→n can be written as:

∇Φ·→n |→s | = Dn(ΦN −ΦP) + Dc

(ΦP2 −Φp1

)(24a)

Dn =|→s |(

→r 1 +

→r 2

)·→n

; Dc = −

(→r 1 +

→r 2

)·→s /|→s |(

→r 1 +

→r 2

)·→n

(24b)

In the above,→r 1 is the distance vector from P to C and

→r 2 is from C to N. The normal and cross

diffusion coefficients, Dn and Dc, at each polygon side involve only geometric variables; they arecalculated only once in the beginning of the computation.

Calculation of a variable, say Y, at the center C of a polygon side is discussed next. This is aninterpolation operation used frequently for variables. A second-order accurate expression is derivedbelow. As shown in Figure 1, a point I is defined as the intercept point between line PN and line P1P2.

A second-order interpolation for point I may be derived to be:

YI =δ1YN + δ2YP

δ1 + δ2(25)

With δ1 =→r 1·→n and δ2 =

→r 2·→n . YI may be used to approximate the value at the side center C.

This treatment, however, does not guarantee second-order accuracy unless→r 1 and

→r 2 are parallel.

A truly second-order expression is derived to be:

YC = YI −Cside(YP2 −YP1) (26a)

Cside =

(δ1→r 2 − δ2

→r 1

)·→s

(δ1 + δ2)|→s |

2 (26b)

Water 2020, 12, 1032 9 of 21

Φc in the convective term of Equation (23) adopts the second-order scheme with a dampingterm. It is derived by blending the first-order upwind scheme with the second-order central differencescheme and may be expressed as:

ΦC = ΦCNC + d

(ΦUP

C −ΦCNC

)(27a)

ΦUPC =

12(ΦP + ΦN) +

12

sign(VC)(ΦP −ΦN) (27b)

In the above, ΦCNC is the second-order interpolation scheme, and d defines the amount of damping

used. In most applications, d = 0.2 ~ 0.3 may be used.With expressions for the diffusion and convection terms done, the final discretized governing

equation at the cell P may be organized as the following linear equation:

APΦP =∑nb

AnbΦnb + Sdi f f + Sconv + SΦ (28)

where “nb” refers to all neighboring polygons surrounding polygon P. The coefficients in thisequation are:

Anb = ΓCDn + Max(0,−hCVC|

→s |)

(29a)

AP =hn

PA

∆t+

∑nb

Anb (29b)

Sdi f f =hn

PA

∆t+

∑all−sides

ΓCDc(ΦP2 −ΦP1) (29c)

Sdi f f =hn

PA

∆t+

∑all−sides

ΓCDc(ΦP2 −ΦP1) (29d)

Sconv =∑

all−sides

(hCVc|

→s |){(1− d)

[δ1

δ1 + δ2−

1− sign(Vc)

2

](ΦN −ΦP)

}−

∑all−sides

(hCVc|

→s |)[(1− d)Cside(ΦP2 −ΦP1)]

Other sediment equations such as the bed elevation equation and the bed dynamics equationsmay be discretized similarly. In terms of time integration, the fraction step method of Yanenko [51] isused. The procedure is as follows:

(hC)int− (hC)n

∆t+∂ cos(α)Vt(hC)int

∂x+∂sin(α)Vt(hC)int

∂y= 0 (30a)

(hC)n+1− (hC)int

∆t=

q∗t −Vt(hC)n+1

Lb(30b)

The advection Equation (30a) is solved implicitly to obtain an intermediate solution (hC)int withknown values at time level n; the initial value problem of (30b) is solved analytically to obtain the newsolution (hC)n+1 at time level (n + 1). The solutions of the bed elevation equation and bed dynamicsequations are relatively straightforward and details are not presented.

4. Model Verification and Validation

Five cases are selected to verify and validate the new sediment transport mobile-bed modeland results are described below. The flow module has been verified and validated before [32]; onlysediment cases are presented.

Water 2020, 12, 1032 10 of 21

4.1. Aggradation in a Straight Channel

Aggradation in alluvial streams may occur due to a variety of reasons. A common scenario is theoversupply of incoming sediments above the stream transport capacity. This may happen in the fieldafter, e.g., heavy precipitation in a large tributary area. The case of the flume experiment of Soni [52] isselected to verify the aggradation simulation capability with the new model.

The case consists of a flat plate bed with a length of 30 m, width of 0.2 m, and slope of 0.00427.The bed is covered with sands with a medium diameter (d50) of 0.32 mm, sand depth of 0.15 m,and specific gravity of 2.65. The case has a constant flow unit discharge of 0.0355 m2/s, average velocityof 0.493 m/s, and average water depth of 0.072 m. Equilibrium of flow and sediment transport isestablished first by simulating long enough in time so that the upstream equilibrium sediment rate isbalanced by the transport capability. Excessive sediment is then added suddenly from the upstream.The Manning’s roughness coefficient is 0.02294 in order to establish the flow equilibrium given the bedslope and flow velocity. Aggradation process is initiated immediately after a sudden increase of thesediment supply rate above the capacity. The excess sediment supply is 0.9qseq with qseq the sedimenttransport capacity. Bedload transport mode was observed in the flume experiment and is used forthe simulation.

A 30-by-5 mesh is used, covering the 30 m by 0.2 m flume bed. The flow is 1D in nature so thenumber of cells in the lateral direction is not important. The time step used is 1.0 s. Further refinementof the mesh or reduction in time step does not change the results more than a few percentages.The simulation is first carried out for 100 min using the upstream sediment supply being qseq; this waythe equilibrium flow and sediment transport is established (no net sediment exchanges between watercolumn and bed). Aggradation simulation then starts with 1.9 times the sediment capacity.

A comparison of the simulated and measured bed elevation changes in time is shown in Figure 2a.Overall agreement is fair, but the model under-predicts the aggradation at a late time (e.g., at 90 min).This may be due to the high uncertainty of the measured data. Soni [52] reported that up to 15% oferrors existed in the rate of sediment addition at the upstream. Another run is made by using a highersediment supply rate to check the model sensitivity to the supply rate. The rate is increased by 15%and the results are recomputed as shown in Figure 2b. A much better agreement is obtained. In thepresent modeling the Engelund-Hansen capacity equation is adopted.

10 of 19

The case consists of a flat plate bed with a length of 30 m, width of 0.2 m, and slope of 0.00427. 407 The bed is covered with sands with a medium diameter (d50) of 0.32 mm, sand depth of 0.15 m, and 408 specific gravity of 2.65. The case has a constant flow unit discharge of 0.0355 m2/s, average velocity 409 of 0.493 m/s, and average water depth of 0.072 m. Equilibrium of flow and sediment transport is 410 established first by simulating long enough in time so that the upstream equilibrium sediment rate is 411 balanced by the transport capability. Excessive sediment is then added suddenly from the upstream. 412 The Manning’s roughness coefficient is 0.02294 in order to establish the flow equilibrium given the 413 bed slope and flow velocity. Aggradation process is initiated immediately after a sudden increase of 414

the sediment supply rate above the capacity. The excess sediment supply is with the 415

sediment transport capacity. Bedload transport mode was observed in the flume experiment and is 416 used for the simulation. 417

A 30-by-5 mesh is used, covering the 30 m by 0.2 m flume bed. The flow is 1D in nature so the 418 number of cells in the lateral direction is not important. The time step used is 1.0 second. Further 419 refinement of the mesh or reduction in time step does not change the results more than a few 420 percentages. The simulation is first carried out for 100 minutes using the upstream sediment supply 421

being ; this way the equilibrium flow and sediment transport is established (no net sediment 422

exchanges between water column and bed). Aggradation simulation then starts with 1.9 times the 423 sediment capacity. 424

A comparison of the simulated and measured bed elevation changes in time is shown in Figure 425 2a. Overall agreement is fair, but the model under-predicts the aggradation at a late time (e.g., at 90 426 minutes). This may be due to the high uncertainty of the measured data. Soni [52] reported that up 427 to 15% of errors existed in the rate of sediment addition at the upstream. Another run is made by 428 using a higher sediment supply rate to check the model sensitivity to the supply rate. The rate is 429 increased by 15% and the results are recomputed as shown in Figure 2b. A much better agreement is 430 obtained. In the present modeling the Engelund-Hansen capacity equation is adopted. 431

(a) 1.9 Supply Rate

(b) 15% More Supply

Figure 2. Comparison of bed elevation changes in time between model prediction and flume data for 432 the aggradation case of Soni [52]). (a) upstream, sediment rate is 1.9 the capacity; (b) upstream 433 sediment supply is 15% more than (a). 434

4.2. Erosion in a Straight Channel 435

Channel erosion and bed armoring occur in many situations such as downstream of a dam. They 436 represent an important class of alluvial processes. Herein the flume experiment of Ashida and 437 Michiue [53] is selected to verify the erosion modeling capability of the new model. 438

The flume used in the experiment was rectangular; it had the width of 0.8 m, length of 20 m, and 439 bed slope of 0.01. The flume bed was filled with non-uniform sediments - a mixture of sands and fine 440 gravels ranging from 0.2 to 10.0 mm in size. The sediment mixture had a medium diameter of 1.5 mm 441

seqq9.0 seqq

seqq

Figure 2. Comparison of bed elevation changes in time between model prediction and flume datafor the aggradation case of Soni [52]). (a) upstream, sediment rate is 1.9 the capacity; (b) upstreamsediment supply is 15% more than (a).

Water 2020, 12, 1032 11 of 21

4.2. Erosion in a Straight Channel

Channel erosion and bed armoring occur in many situations such as downstream of a dam.They represent an important class of alluvial processes. Herein the flume experiment of Ashida andMichiue [53] is selected to verify the erosion modeling capability of the new model.

The flume used in the experiment was rectangular; it had the width of 0.8 m, length of 20 m, andbed slope of 0.01. The flume bed was filled with non-uniform sediments—a mixture of sands and finegravels ranging from 0.2 to 10.0 mm in size. The sediment mixture had a medium diameter of 1.5 mmand a standard deviation of 3.47 (Figure 3 shows the initial bed gradation). The simulated case has aconstant clear water flow of 0.0314 m3/s, an average velocity of 0.654 m/s, and water depth of 6.0 mmat the downstream boundary.

12 of 19

480

Figure 3. Comparison of predicted and measured bed gradation at a location 10-m from the 481 downstream model boundary; measured data are from Ashida-Michiue [53]. 482

483

Figure 4. Comparison of predicted and measured scour depth variation in time at three locations: 13 484 m (red), 10 m (blue), and 7 m (black) from the downstream model boundary; “Measured” = Ashida-485 Michiue [53]; “Wu Grain Stress” = the results obtained with the modified grain shear stress 486 calculation. 487

4.3. Erosion and Depostion in Bends 488

Two bend flows, reported by Struiksma et al. [54], are used to validate the ability of the new 489 model to simulate erosion and deposition in stram bends. One is case T2 with a 140° bend filled with 490 uniform sediments; the other is case T4 with a 108.1° bend filled with non-uniform sediments. The 491 geometry of the two bends is shown in Figure 5. Both were tested at the Delft Hydraulics Laboratory: 492 T2 used the DHL curved flume and T4 used the Waal Bend flume. 493

Case T2 had a length of 29.32 m and a radius of curvature of 12 m (Figure 5); relevant geometry 494 and test conditions are listed in Table 2. The flume bed was initially flat laterally and filled with 495 uniform sediments of 0.45 mm in diameter. The equilibrium bed topography was achieved after long 496 enough water flow over the flume bed. 497

498

Figure 3. Comparison of predicted and measured bed gradation at a location 10-m from the downstreammodel boundary; measured data are from Ashida-Michiue [53].

A 42-by-6 mesh is used for the simulation, covering the 20 m by 0.8 m model domain (channelbed). Note that the lateral mesh number is not important due to the 1D nature of the flow. The timestep is 1.0 second. Further refinement of the mesh in the flow direction or reduction of the time stepdoes not change the solutions more than a few percentages. Twelve sediment size classes are usedto represent the sediment mixture. The range of size classes and the corresponding initial fractionalgradations on the bed are in Table 1; the initial bed gradation is also plotted in Figure 3. The Manning’sroughness coefficient is 0.025; it is based on the flow calibration to achieve flow equilibrium withthe given slope and velocity. For a mixed sand-gravel bed, the Parker sediment capacity equation isused with the default constants (i.e., θC = 0.04 and α = 0.65). At time zero, clear water flows into thechannel; afterwards, the degradation process is initiated. Only bedload transport was observed in theexperiment and is thus simulated.

Table 1. Sediment size classes and the initial fractional content (gradation) on bed.

DiameterRange(mm)

0.2–0.3

0.3–0.4

0.4–0.6

0.6–0.8

0.8–1.0

1.0–1.5

1.5–2.0

2.0–3.0

3.0–4.0

4.0–6.0

6.0–8.0

8.0–10.

Content(%) 7.45 12.4 15.9 4.4 3.6 6.79 4.0 9.18 10.2 18.1 6.0 2.0

The experimental data showed that erosion started immediately once flow enters the flume andscour depth increased quickly for the first 100 min. Afterwards, erosion slowed down and an armoringlayer was formed. The model-predicted, armored bed gradation is shown in Figure 3 at a location of

Water 2020, 12, 1032 12 of 21

10 m from the downstream boundary. A comparison with the measured gradation at the same locationshows that the agreement is relatively good, indicating the ability of the new model to predict thearmoring process.

The erosion process is also compared between the model and the flume data in Figure 4 at threelocations. The prediction of the time-varying scour process is less satisfactory, but the final maximumscour is predicted well. In view of a better prediction reported by Wu [16], we had tried to find out thecause. According to the discussion in [16], along with a personal communication with Dr. Wu, the causewas attributed to the bedform change while the scour was developing: the bedform was changing froma flat bed to a fully developed bed in the flume experiment. In the modeling of Wu [16], the bedformchange was taken into account by adopting a time-dependent variation of the bed grain shear stress.The present modeling, however, used a constant. In order to see the impact of the time-varyingbedform, the same functional form of the grain shear stress used by [16] is implemented; the model isrun again without changing other model inputs. The new predicted scour results are shown in Figure 4as dashed lines, designated as “Predicted: Wu Grain Stress.” It is seen that the variable grain stressprocedure indeed improves the agreement between the model prediction and measured data, and thenew results are close to Wu [16]. Since the grain stress changing procedure used is not general; it isnot implemented as a feature in the new model. We prefer to take the results without the treatment(in solid lines) as the model prediction. The study, however, points to potential uncertainty in modelpredictions when bedform changes. Future studies are needed on how to incorporate a more generalprocedure so that bedform changes may be taken into accounts.

1

Figure 4. Comparison of predicted and measured scour depth variation in time at threelocations: 13 m (red), 10 m (blue), and 7 m (black) from the downstream model boundary;“Measured” = Ashida-Michiue [53]; “Wu Grain Stress” = the results obtained with the modifiedgrain shear stress calculation.

4.3. Erosion and Depostion in Bends

Two bend flows, reported by Struiksma et al. [54], are used to validate the ability of the new modelto simulate erosion and deposition in stram bends. One is case T2 with a 140◦ bend filled with uniformsediments; the other is case T4 with a 108.1◦ bend filled with non-uniform sediments. The geometry ofthe two bends is shown in Figure 5. Both were tested at the Delft Hydraulics Laboratory: T2 used theDHL curved flume and T4 used the Waal Bend flume.

Water 2020, 12, 1032 13 of 21 13 of 19

(a) Case T2

(b) Case T4

Figure 5. Bend geometry for (a) Cases T2 and (b) Case T4 studied by Struiksma et al. [54]. 499

Table 2. Flume geometry and experiment conditions for case T2 of Struiksma et al. [54]. 500

Flume Width

(m)

Discharge

(m3/s)

Water Depth

(m)

Velocity

(m/s)

Bed

Slope

Froude

Number

d50

(mm)

Manning’s

Coefficient

1.5 0.062 0.10 0.41 0.203% 0.41 0.45 0.023

A total of 960 mesh cells are used to cover the mode domain. Further mesh refinement has not 501 changed the results more than 2%. An unsteady simulation is carried out with a time step of 10 502 seconds. At the upstream (x = 0), a flow discharge of 0.062 m3/s is imposed, and the sediment supply 503 rate is estimated with the sediment capacity equation. At the exit, the water surface elevation of 0.1 504 m is maintained. The Engelund-Hansen sediment capacity equation is used and the beadload 505 transport mode is adopted based on the laboratory observation. The equilibrium bedform is reached 506 after a sufficient time (about 10 hours) and the computed bed elevation is shown in Figure 6. In 507 addition, a comparison of the computed and measured water depths along two profiles, 0.375 m from 508 the inner and the outer banks, is shown in Figure 7. 509

510

Figure 6. Computed equilibrium bed elevation for case T2 of Struiksma et al. [54]. 511

Figure 5. Bend geometry for (a) Cases T2 and (b) Case T4 studied by Struiksma et al. [54].

Case T2 had a length of 29.32 m and a radius of curvature of 12 m (Figure 5); relevant geometryand test conditions are listed in Table 2. The flume bed was initially flat laterally and filled withuniform sediments of 0.45 mm in diameter. The equilibrium bed topography was achieved after longenough water flow over the flume bed.

Table 2. Flume geometry and experiment conditions for case T2 of Struiksma et al. [54].

FlumeWidth (m)

Discharge(m3/s)

WaterDepth (m)

Velocity(m/s)

BedSlope

FroudeNumber

d50(mm)

Manning’sCoefficient

1.5 0.062 0.10 0.41 0.203% 0.41 0.45 0.023

A total of 960 mesh cells are used to cover the mode domain. Further mesh refinement has notchanged the results more than 2%. An unsteady simulation is carried out with a time step of 10 seconds.At the upstream (x = 0), a flow discharge of 0.062 m3/s is imposed, and the sediment supply rate isestimated with the sediment capacity equation. At the exit, the water surface elevation of 0.1 m ismaintained. The Engelund-Hansen sediment capacity equation is used and the beadload transportmode is adopted based on the laboratory observation. The equilibrium bedform is reached after asufficient time (about 10 hours) and the computed bed elevation is shown in Figure 6. In addition,a comparison of the computed and measured water depths along two profiles, 0.375 m from the innerand the outer banks, is shown in Figure 7.

The comparisons show that the new model predicts the pool and bar formation adequatelyalthough the initial bed is flat laterally. A bar is formed at the inner bank while a pool occurs at theouter bank. The agreement between the computation and measurements is satisfactory.

Next, case T4 is simulated: it consisted of a bend with a circular arc of 108.1o turn and 41.5 m inlength. Both the entrance and exit of the bend have a section of the straight channel attached. The radiusof the arc from the centerline was 22 m. Initial bed had a slope of 0.128% longitudinally but flat laterally.The bed was covered with non-uniform mixtures having d50 = 0.6 mm. Some characteristic parametersof the case are listed in Table 3.

Water 2020, 12, 1032 14 of 21

13 of 19

(a) Case T2

(b) Case T4

Figure 5. Bend geometry for (a) Cases T2 and (b) Case T4 studied by Struiksma et al. [54]. 499

Table 2. Flume geometry and experiment conditions for case T2 of Struiksma et al. [54]. 500

Flume Width

(m)

Discharge

(m3/s)

Water Depth

(m)

Velocity

(m/s)

Bed

Slope

Froude

Number

d50

(mm)

Manning’s

Coefficient

1.5 0.062 0.10 0.41 0.203% 0.41 0.45 0.023

A total of 960 mesh cells are used to cover the mode domain. Further mesh refinement has not 501 changed the results more than 2%. An unsteady simulation is carried out with a time step of 10 502 seconds. At the upstream (x = 0), a flow discharge of 0.062 m3/s is imposed, and the sediment supply 503 rate is estimated with the sediment capacity equation. At the exit, the water surface elevation of 0.1 504 m is maintained. The Engelund-Hansen sediment capacity equation is used and the beadload 505 transport mode is adopted based on the laboratory observation. The equilibrium bedform is reached 506 after a sufficient time (about 10 hours) and the computed bed elevation is shown in Figure 6. In 507 addition, a comparison of the computed and measured water depths along two profiles, 0.375 m from 508 the inner and the outer banks, is shown in Figure 7. 509

510

Figure 6. Computed equilibrium bed elevation for case T2 of Struiksma et al. [54]. 511 Figure 6. Computed equilibrium bed elevation for case T2 of Struiksma et al. [54]. 14 of 19

512

Figure 7. Comparison of the computed and measured water depths along lines 0.375 m from inner 513 and outer banks for case T2 of Struiksma et al. [54]. 514

The comparisons show that the new model predicts the pool and bar formation adequately 515 although the initial bed is flat laterally. A bar is formed at the inner bank while a pool occurs at the 516 outer bank. The agreement between the computation and measurements is satisfactory. 517

Next, case T4 is simulated: it consisted of a bend with a circular arc of 108.1o turn and 41.5 m in 518 length. Both the entrance and exit of the bend have a section of the straight channel attached. The 519 radius of the arc from the centerline was 22 m. Initial bed had a slope of 0.128% longitudinally but 520

flat laterally. The bed was covered with non-uniform mixtures having mm. Some 521

characteristic parameters of the case are listed in Table 3. 522 The 2D mesh consists of 1,824 cells which is found sufficient as further refinement has not 523

changed the results more than a few percentages. The sediment mixture is divided into four size 524 classes: 15.9% of fine sand (0.125 to 0.26 mm), 34.1% medium sand (0.26 to 0.6 mm), 34.1% coarse 525 sand (0.6 to 1.38 mm), and 15.9% vary coarse sand (1.38 to 2.0 mm). A truly unsteady simulation is 526 carried out with a time step of 10 seconds. The bed is initially filled with the sediment mixture. At 527 the upstream (x = 0), a flow discharge of 0.121 m3/s is imposed and the sediment supply is calculated 528 with the sediment capacity equation. At the exit, the water surface elevation is maintained at 0.12 m. 529

The equilibrium bedform is obtained after about 10 hours and plotted in Figure 8. A comparison 530

of the computed and measured equilibrium water depth and medium sediment diameter ( ) is 531

shown in Figure 9 along two profiles: 0.11B from the inner and outer banks (B = 2.3 m is the channel 532 width). A comparison with the measured data shows that the new model is capable of predicting the 533 pool and bar formation as well as the sediment sorting process. Overall agreement between the model 534 and measured data is good. 535

Table 3. Flume geometry and experiment conditions for case T4 of Struiksma et al. [54]. 536

Flume Width

(m)

Discharge

(m3/s)

Water Depth

(m)

Velocity

(m/s)

Bed

Slope

Froude

Number

d50

(mm)

Manning’s

Coefficient

2.3 0.121 0.12 0.44 0.128% 0.41 0.60 0.02

6.050 d

50d

Figure 7. Comparison of the computed and measured water depths along lines 0.375 m from inner andouter banks for case T2 of Struiksma et al. [54].

Table 3. Flume geometry and experiment conditions for case T4 of Struiksma et al. [54].

FlumeWidth (m)

Discharge(m3/s)

WaterDepth (m)

Velocity(m/s)

BedSlope

FroudeNumber

d50(mm)

Manning’sCoefficient

2.3 0.121 0.12 0.44 0.128% 0.41 0.60 0.02

The 2D mesh consists of 1824 cells which is found sufficient as further refinement has not changedthe results more than a few percentages. The sediment mixture is divided into four size classes: 15.9%of fine sand (0.125 to 0.26 mm), 34.1% medium sand (0.26 to 0.6 mm), 34.1% coarse sand (0.6 to 1.38mm), and 15.9% vary coarse sand (1.38 to 2.0 mm). A truly unsteady simulation is carried out with atime step of 10 seconds. The bed is initially filled with the sediment mixture. At the upstream (x = 0),a flow discharge of 0.121 m3/s is imposed and the sediment supply is calculated with the sedimentcapacity equation. At the exit, the water surface elevation is maintained at 0.12 m.

The equilibrium bedform is obtained after about 10 hours and plotted in Figure 8. A comparisonof the computed and measured equilibrium water depth and medium sediment diameter (d50) isshown in Figure 9 along two profiles: 0.11B from the inner and outer banks (B = 2.3 m is the channelwidth). A comparison with the measured data shows that the new model is capable of predicting the

Water 2020, 12, 1032 15 of 21

pool and bar formation as well as the sediment sorting process. Overall agreement between the modeland measured data is good.

15 of 19

537

Figure 8. Computed equilibrium bed elevation for case T4 of Struiksma et al. [54]. 538

Figure 9. Comparison of computed and measured water depth (left) and (right) along the lines 539

0.11B from the inner and outer banks for case T4 of Struiksma et al. [54]. 540

4.4. Alternating Bar Formation Downstream of an Inserted Dike 541

Alternate bar formation study was carried out at the Delft Hydraulics Laboratory by Struiksma 542 and Crosato [35]. When a dike was inserted into a straight channel, forced alternate bars were formed 543 downstream. The flume experiment started with a straight channel 0.6 m in width and 0.3% in slope 544 under a well-defined constant flow. At the upstream, a plate (dike) was inserted to restrict the inflow 545 section. The channel bed was initially flat and covered with almost uniform fine sediments with a 546 median diameter (d50) of 0.216 mm. When the bed reached equilibrium, bed topography was 547 measured which is available for numerical model verification. The experimental conditions are 548 shown in Table 4. 549

Table 4. Key flume geometry and experiment conditions with the Struiksma and Crosato [35] case. 550

Flume

Width (m)

Discharge

(m3/s)

Water

Depth (m)

Velocity

(m/s)

Bed

Slope

Froude

Number

Manning’s

Coefficient

0.60 0.00685 0.044 0.26 0.3% 0.39 0.0263

The numerical model domain and the mesh are shown in Figure 10. The mesh consists of 236-551 by-24 cells; it is sufficient as a further refinement has not changed the results by more than a few 552 percentages. An unsteady simulation is carried out, with a time step of 5 seconds, until equilibrium 553

50d

Figure 8. Computed equilibrium bed elevation for case T4 of Struiksma et al. [54].

15 of 19

537

Figure 8. Computed equilibrium bed elevation for case T4 of Struiksma et al. [54]. 538

Figure 9. Comparison of computed and measured water depth (left) and (right) along the lines 539

0.11B from the inner and outer banks for case T4 of Struiksma et al. [54]. 540

4.4. Alternating Bar Formation Downstream of an Inserted Dike 541

Alternate bar formation study was carried out at the Delft Hydraulics Laboratory by Struiksma 542 and Crosato [35]. When a dike was inserted into a straight channel, forced alternate bars were formed 543 downstream. The flume experiment started with a straight channel 0.6 m in width and 0.3% in slope 544 under a well-defined constant flow. At the upstream, a plate (dike) was inserted to restrict the inflow 545 section. The channel bed was initially flat and covered with almost uniform fine sediments with a 546 median diameter (d50) of 0.216 mm. When the bed reached equilibrium, bed topography was 547 measured which is available for numerical model verification. The experimental conditions are 548 shown in Table 4. 549

Table 4. Key flume geometry and experiment conditions with the Struiksma and Crosato [35] case. 550

Flume

Width (m)

Discharge

(m3/s)

Water

Depth (m)

Velocity

(m/s)

Bed

Slope

Froude

Number

Manning’s

Coefficient

0.60 0.00685 0.044 0.26 0.3% 0.39 0.0263

The numerical model domain and the mesh are shown in Figure 10. The mesh consists of 236-551 by-24 cells; it is sufficient as a further refinement has not changed the results by more than a few 552 percentages. An unsteady simulation is carried out, with a time step of 5 seconds, until equilibrium 553

50dFigure 9. Comparison of computed and measured water depth (left) and d50 (right) along the lines0.11B from the inner and outer banks for case T4 of Struiksma et al. [54].

4.4. Alternating Bar Formation Downstream of an Inserted Dike

Alternate bar formation study was carried out at the Delft Hydraulics Laboratory by Struiksmaand Crosato [35]. When a dike was inserted into a straight channel, forced alternate bars were formeddownstream. The flume experiment started with a straight channel 0.6 m in width and 0.3% in slopeunder a well-defined constant flow. At the upstream, a plate (dike) was inserted to restrict the inflowsection. The channel bed was initially flat and covered with almost uniform fine sediments with amedian diameter (d50) of 0.216 mm. When the bed reached equilibrium, bed topography was measuredwhich is available for numerical model verification. The experimental conditions are shown in Table 4.

Table 4. Key flume geometry and experiment conditions with the Struiksma and Crosato [35] case.

FlumeWidth (m)

Discharge(m3/s)

WaterDepth (m)

Velocity(m/s)

BedSlope

FroudeNumber

Manning’sCoefficient

0.60 0.00685 0.044 0.26 0.3% 0.39 0.0263

The numerical model domain and the mesh are shown in Figure 10. The mesh consists of 236-by-24cells; it is sufficient as a further refinement has not changed the results by more than a few percentages.An unsteady simulation is carried out, with a time step of 5 seconds, until equilibrium bed topography

Water 2020, 12, 1032 16 of 21

is reached. At the upstream (x = −15.0 m), a constant discharge of 0.00685 m3/s is imposed and thesediment supply rate is based on the sediment transport capacity as equilibrium solution is sought.At the downstream (x = 25 m), a water depth of 0.044 m is specified. The Parker capacity equationis used.

16 of 19

bed topography is reached. At the upstream (x = −15.0 m), a constant discharge of 0.00685 m3/s is 554 imposed and the sediment supply rate is based on the sediment transport capacity as equilibrium 555 solution is sought. At the downstream (x = 25 m), a water depth of 0.044 m is specified. The Parker 556 capacity equation is used. 557

The predicted equilibrium bed topography, in the form of net erosion and deposition, is 558 displayed in Figure 11; a comparison of the equilibrium bed profile between the model and measured 559 data is in Figure 12 along a line 0.1 m from the bottom boundary. It is seen that a series of alternating 560 bars are developed downstream of the dike. The bar and pool depths (amplitude) are, on average, 561 about 25% of the average water depth, and the amplitude is mildly damped downstream. The 562 average bar wavelength is approximately eleven times the channel width, much larger than the 563 typical downstream migrating free bars. Results in Figure 12 show that the comparison between the 564 prediction and measured data is good. The amplitude of the alternate bars is predicted satisfactorily 565 while the wavelength is slightly over-predicted. The results demonstrate that the fully non-linear 566 numerical models such as the present model is capable of predicting the alternate bar development 567 as a response to disturbances introduced into a stream. 568

569

Figure 10. Channel geometry, dimension, and 2D mesh for the Struiksma and Crosato [35] case. 570

571

Figure 11. Predicted deposit depth (positive) and erosion depth (negative). 572

573

Figure 10. Channel geometry, dimension, and 2D mesh for the Struiksma and Crosato [35] case.

The predicted equilibrium bed topography, in the form of net erosion and deposition, is displayedin Figure 11; a comparison of the equilibrium bed profile between the model and measured data isin Figure 12 along a line 0.1 m from the bottom boundary. It is seen that a series of alternating barsare developed downstream of the dike. The bar and pool depths (amplitude) are, on average, about25% of the average water depth, and the amplitude is mildly damped downstream. The average barwavelength is approximately eleven times the channel width, much larger than the typical downstreammigrating free bars. Results in Figure 12 show that the comparison between the prediction andmeasured data is good. The amplitude of the alternate bars is predicted satisfactorily while thewavelength is slightly over-predicted. The results demonstrate that the fully non-linear numericalmodels such as the present model is capable of predicting the alternate bar development as a responseto disturbances introduced into a stream.

16 of 19

bed topography is reached. At the upstream (x = −15.0 m), a constant discharge of 0.00685 m3/s is 554 imposed and the sediment supply rate is based on the sediment transport capacity as equilibrium 555 solution is sought. At the downstream (x = 25 m), a water depth of 0.044 m is specified. The Parker 556 capacity equation is used. 557

The predicted equilibrium bed topography, in the form of net erosion and deposition, is 558 displayed in Figure 11; a comparison of the equilibrium bed profile between the model and measured 559 data is in Figure 12 along a line 0.1 m from the bottom boundary. It is seen that a series of alternating 560 bars are developed downstream of the dike. The bar and pool depths (amplitude) are, on average, 561 about 25% of the average water depth, and the amplitude is mildly damped downstream. The 562 average bar wavelength is approximately eleven times the channel width, much larger than the 563 typical downstream migrating free bars. Results in Figure 12 show that the comparison between the 564 prediction and measured data is good. The amplitude of the alternate bars is predicted satisfactorily 565 while the wavelength is slightly over-predicted. The results demonstrate that the fully non-linear 566 numerical models such as the present model is capable of predicting the alternate bar development 567 as a response to disturbances introduced into a stream. 568

569

Figure 10. Channel geometry, dimension, and 2D mesh for the Struiksma and Crosato [35] case. 570

571

Figure 11. Predicted deposit depth (positive) and erosion depth (negative). 572

573

Figure 11. Predicted deposit depth (positive) and erosion depth (negative).

16 of 19

bed topography is reached. At the upstream (x = −15.0 m), a constant discharge of 0.00685 m3/s is 554 imposed and the sediment supply rate is based on the sediment transport capacity as equilibrium 555 solution is sought. At the downstream (x = 25 m), a water depth of 0.044 m is specified. The Parker 556 capacity equation is used. 557

The predicted equilibrium bed topography, in the form of net erosion and deposition, is 558 displayed in Figure 11; a comparison of the equilibrium bed profile between the model and measured 559 data is in Figure 12 along a line 0.1 m from the bottom boundary. It is seen that a series of alternating 560 bars are developed downstream of the dike. The bar and pool depths (amplitude) are, on average, 561 about 25% of the average water depth, and the amplitude is mildly damped downstream. The 562 average bar wavelength is approximately eleven times the channel width, much larger than the 563 typical downstream migrating free bars. Results in Figure 12 show that the comparison between the 564 prediction and measured data is good. The amplitude of the alternate bars is predicted satisfactorily 565 while the wavelength is slightly over-predicted. The results demonstrate that the fully non-linear 566 numerical models such as the present model is capable of predicting the alternate bar development 567 as a response to disturbances introduced into a stream. 568

569

Figure 10. Channel geometry, dimension, and 2D mesh for the Struiksma and Crosato [35] case. 570

571

Figure 11. Predicted deposit depth (positive) and erosion depth (negative). 572

573

Figure 12. Comparison of predicted and measured erosion and deposition depths along a straight lineof y = 0.1 m; depth is normalized with the average water depth of 0.044 m.

Water 2020, 12, 1032 17 of 21

4.5. Erosion and Deposition on a Section of the Middle Rio Grande

Finally, the new model is applied to a section of the Middle Rio Grande (MRG) for validation andapplication—the Bosque del Apache National Wildlife Refuge (BDANWR) area. BDANWR is about11 miles upstream of the Tiffany Junction. In May 2008, BDANWR experienced a plug formation—asudden and large sediment deposition which blocked the flows in the main river. This section hasbeen subject to extensive field studies since to understand the erosion and deposition processes after apilot channel was dug to reconnect the main river [55,56]. Measured bed elevation changes on manycross-sections (Figure 13a) were available in both 2008 and 2009 so modeling results may be comparedand validated. 18 of 19

(a) Measurement XS

(b) 2D Mesh

Figure 13. (a) Cross sections (XS) of the field measurement. (b) The 2D mesh for the numerical 607 modeling (flow is from top to bottom, or North to South). 608

(a) SO-1525

(b) SO-1530.5

(c) SO-1544

(d) SO-1550

Figure 14. Comparison of predicted and measured bed elevation changes at four cross sections. 609

Figure 13. (a) Cross sections (XS) of the field measurement. (b) The 2D mesh for the numerical modeling(flow is from top to bottom, or North to South).

The simulated MRG section has the upstream at the North Boundary of BDANWR near SO-1513.5(river mile 84) and the downstream boundary near SO-1564.4 (river mile 78.7)—see Figure 13a for thelocations. The solution domain covers about 5.3 miles longitudinally. The 2D mesh consists of mixedquadrilaterals and triangles with a total of 10,865 cells (Figure 13b).

Other model inputs are as follows. The Manning’s roughness coefficient is based on the 1Dmodeling studies by Borough [57] and Collin [58] without changes. Bed sediment gradation prior tothe plug formation is based on the data collected by Bauer [59] in June and July 2006. The data showedthat about 99.5% of the sediments ranged from 0.0625 to 16 mm in diameter with d50 = 0.33 mm. In thenumerical modeling, seven size classes are used to represent all sediments (0.0625–16 mm) in thesystem. An unsteady simulation is carried out for the time period from the beginning of November2008 to the end of July 2009. The flow discharge at the upstream is based on the daily mean flowmeasured at the USGS San Acacia station (#08354900), while the sediment input is based on the ratingcurve developed by Collins [58]. The Engelund-Hansen equation is used for the sediment capacity.

Water 2020, 12, 1032 18 of 21

The erosion and deposition of the study reach in the period of 1 November 2008 to 31 July 2009 aresimulated with the new model. The predicted channel bed elevation changes are compared with thecross-section data measured in July 2009. Comparisons at four cross sections are displayed in Figure 14.Overall, the reach is mostly erosional during the simulation period, which is expected as the plugformed in May 20018 was being eroded after the pilot channel was dug in October 2008. The agreementbetween the predicted and measured erosion is relatively good except for a small section betweenSO-1539 and SO-1544. At SO-1544, e.g., incising is predicted, not widening. This is due to that the newmodel does not simulate bank erosion so channel widening is not predicted.

18 of 19

(a) Measurement XS

(b) 2D Mesh

Figure 13. (a) Cross sections (XS) of the field measurement. (b) The 2D mesh for the numerical 607 modeling (flow is from top to bottom, or North to South). 608

(a) SO-1525

(b) SO-1530.5

(c) SO-1544

(d) SO-1550

Figure 14. Comparison of predicted and measured bed elevation changes at four cross sections. 609 Figure 14. Comparison of predicted and measured bed elevation changes at four cross sections.

5. Conclusions

A new depth-averaged 2D sediment transport mobile-bed model is developed, verified andvalidated. New contributions of the proposed model include: (a) polygon-based mesh; (b) a singlesediment transport equation simulating suspended load, bedload and mixed load simultaneously;(c) truly unsteady, tightly-coupled modeling among flow, sediment transport and bed dynamicsmodules; and (d) a generally applicable formulation for multi-size, non-equilibrium sediment transport.The model is named SRH-2D and has been freely released to the public.

There are other features of the new model. For example, multiple subsurface bed layers are allowedso that bed stratigraphy may be taken into consideration in erosion simulation. The finite-volumemethod is adopted such that mass conservations of both water and sediment are satisfied locally andglobally. Implicit time integration is used so that the solution process is robust and stable.

In this paper, four flume cases are selected to verify the new model, and the erosion and depositionof a section of the Middle Rio Grande is used as a validation and demonstration case. Model resultshave been compared with the available measured data. It is found that the agreement has been goodfor all cases.

Funding: The research was funded by the Science and Technology Office, U.S. Bureau of Reclamation and theWater Resources Agency, Taiwan.

Conflicts of Interest: The authors declare no conflict of interest.

Water 2020, 12, 1032 19 of 21

References

1. ASCE. Earthen embankment breaching. J. Hydraul. Eng. 2011, 137, 1549–1564. [CrossRef]2. Gibson, S.; Brunner, G.; Piper, S.; Jensen, M. Sediment Transport Computations in HEC-RAS. In Proceedings

of the Eighth Federal Interagency Sedimentation Conference (8th FISC), Reno, NV, USA, 2–6 April 2006;pp. 57–64.

3. USACE. New One-Dimensional Sediment Features in HEC-RAS 5.0 and 5.1. 2018. Availableonline: https://www.researchgate.net/publication/317073446_New_One-imensional_Sediment_Features_in_HEC-RAS_50_and_51 (accessed on 10 January 2020).

4. DHI. MIKE 11. In Users Manual; Danish Hydraulic Institute: Horsholm, Denmark, 2005.5. Wu, W.; Vieira, D.A. One-Dimensional Channel Network Model CCHE1D Version 3.0.—Technical Manual; Technical

Report No. NCCHE-TR-2002-1; National Center for Computational Hydroscience and Engineering, TheUniversity of Mississippi: Oxford, MS, USA, 2002.

6. Huang, J.; Greimann, B.P. GSTAR-1D, General Sediment Transport for Alluvial Rivers—One Dimension; TechnicalService Center, U.S. Bureau of Reclamation: Denver, CO, USA, 2007.

7. FHWA. Evaluating Scour at Bridges, HEC-18, 5th ed.; Report, No. HIF-FHWA-12-003; Arneson, L.A.,Zevenbergen, L.W., Lagasse, P.F., Clopper, P.E., Eds.; U.S. Department of Transportation, Federal HighwayAdministration: Washington, DC, USA, 2012.

8. Hogan, S.A. Advancements in Bridge Scour Assessment with 2D Hydraulic Modeling Using SRH-2D/SMS.In Proceedings of the 9th International Conference on Scour and Erosion, Taipei, Taiwan, 5–8 November 2018.

9. Chow, V.T.; Ben-Zvi, A. Hydrodynamic modeling of two-dimensional watershed flow. J. Hydraul. Div. ASCE1973, 99, 2023–2040.

10. Kuipers, J.; Vreugdenhil, C.B. Calculation of Two Dimensional Horizontal Flow; Reports S163, Part 1; DefltHydraulics Lab.: Delft, The Netherlands, 1973.

11. Celik, I.; Rodi, W. Modeling suspended sediment transport in nonequilibrium situations. J. Hydraul. Eng.1998, 114, 1157–1191. [CrossRef]

12. Spasojevic, M.; Holly, F.M., Jr. 2-D bed evolution in natural watercourses—New simulation approach.J. Waterw. Port Coast. Ocean Eng. 1990, 116, 425–443. [CrossRef]

13. Minh Duc, B. Berechnung der Stroemung und des Sedimenttransorts in Fluessen Mit Einem TieffengemitteltenNumerischen Verfahren. Ph.D. Thesis, The University of Karlsruhe, Karlsruhe, Germany, 1998. (In German)

14. Olsen, N.R.B. Two-dimensional numerical modeling of flushing processes in water reservoirs. J. Hydraul.Res. 1999, 37, 3–16. [CrossRef]

15. Wu, W.; Wang, S.S.Y.; Jia, T.; Robinson, K.M. Numerical simulation of two-dimensional headcut migration.In Proceedings of the 1999 International Water Resources Engineering Conference, on CD-ROM., ASCE,Seattle, WA, USA, 8–12 August 1999.

16. Wu, W. Depth-averaged two-dimensional numerical modeling of unsteady flow and non-uniform sedimenttransport in open channels. J. Hydraul. Eng. 2004, 130, 1013–1024. [CrossRef]

17. Hung, M.C.; Hsieh, T.Y.; Wu, C.H.; Yang, J.C. Two-dimensional nonequilibrium noncohesive and cohesivesediment transport model. J. Hydraul. Eng. 2009, 135, 369–382. [CrossRef]

18. Huang, W.; Cao, Z.; Pender, G.; Liu, Q.; Carling, P. Coupled flood and sediment transport modelling withadaptive mesh refinement. Sci. China Technol. Sci. 2015, 58, 1425–1438. [CrossRef]

19. Jia, Y.; Wang, S.S.Y. CCHE2D: Two-Dimensional Hydrodynamic and Sediment Transport Model for Unsteady OpenChannel Flows Over Loose Bed; Technical Report: NCCHETR-2001-01; National Center for ComputationalHydroscience and Engineering, The University of Mississippi: Oxford, MS, USA, 2001.

20. Hervouet, J.-M. Hydrodynamics of Free Surface Flows; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2007;pp. 1–360.

21. Casulli, V.; Walters, R.A. An unstructured grid, three dimensional model based on the shallow waterequations. Int. J. Numer. Methods Fluids 2000, 32, 331–348. [CrossRef]

22. Bever, A.J.; MacWilliams, M.L. Simulating sediment transport processes in San Pablo Bay using coupledhydrodynamic, wave, and sediment transport models. Mar. Geol. 2013, 345, 235–253. [CrossRef]

23. Deltares. Delft3D-FLOW. In Simulation of Multi-Dimensional Hydrodynamic Flow and Transport Phenomena,including Sediments–User Manual; Version 3.04, rev. 11114; Deltares: Delft, The Netherlands, 2010.

Water 2020, 12, 1032 20 of 21

24. Kernkamp, H.W.J.; Van Dam, A.; Stelling, G.S.; De Goede, E.D. Efficient scheme for the shallow waterequations on unstructured grids with application to the Continental Shelf. Ocean Dynam. 2010, 29, 1175–1188.[CrossRef]

25. Achete, F.M.; van der Wegen, M.; Roelvink, D.; Jaffe, B. A 2-D process-based model for suspended sedimentdynamics: A first step towards ecological modeling. Hydrol. Earth Syst. Sci. 2015, 19, 2837–2857. [CrossRef]

26. Deltares. D-Flow Flexible Mesh, Technical Reference Manual; Deltares: Delft, The Netherlands, 2014; 82p.27. Iverson, R.M.; Ouyang, C. Entrainment of bed material by earth-surface mass flows: Review and reformulation

of depth-integrated theory. Rev. Geophys. 2015, 53, 27–58. [CrossRef]28. Cao, Z.; Xia, C.; Pender, G.; Liu, Q. Shallow water hydro-sediment-morphodynamic equations for fluvial

processes. J. Hydraul. Eng. 2017, 143, 02517001. [CrossRef]29. Liu, X.; Infante Sedano, J.A.; Mohammadian, A. A coupled two-dimensional numerical model for rapidly

varying flow, sediment transport and bed morphology. J. Hydraul. Res. 2015, 53, 609–621. [CrossRef]30. Greimann, B.P.; Lai, Y.G.; Huang, J.C. Two-dimensional total sediment load model equations. J. Hydraul. Eng.

2008, 134, 1142–1146. [CrossRef]31. Lai, Y.G. Modeling Stream Bank Erosion: Performance for Practical Streams and Future Needs. Water 2017,

9, 950. [CrossRef]32. Lai, Y.G. Two-Dimensional Depth-Averaged Flow Modeling with an Unstructured Hybrid Mesh. J. Hydraul.

Eng. 2010, 136, 12–23. [CrossRef]33. Rodi, W. Turbulence Models and Their Application in Hydraulics, 3rd ed.; IAHR Monograph: Rotterdam,

The Netherlands, 1993.34. Odgaard, A.J. Transverse bed slope in alluvial channel beds. J. Hydraul. Div. ASCE 1981, 107, 1677–1694.35. Struiksma, N.; Crosato, A. Analysis of a 2D bed topography model for rivers. In River Meandering; Ikeda, S.,

Parker, G., Eds.; AGU Water Resources Monograph 12: Washington, DC, USA, 1989.36. Talmon, A.M.; van Mierlo, M.C.L.M.; Struiksma, N. Laboratory measurements of the direction of sediment

transport on transverse alluvial-bed slopes. J. Hydraul. Res. 1995, 33, 495–517. [CrossRef]37. Holly, F.M.; Rahuel, J.L. New Numerical/Physical Framework for Mobile-Bed Modeling, Part 1: Numerical

and Physical Principles. J. Hydraul. Res. 1990, 28, 401–416. [CrossRef]38. Engelund, F.; Hansen, E. A Monograph on Sediment Transport in Alluvial Streams; Teknish Forlag, Technical

Press: Copenhagen, Denmark, 1972.39. Parker, G. Surface-based bedload transport relation for gravel rivers. J. Hydraul. Res. 1990, 28, 417–428.

[CrossRef]40. Andrews, E.D. Bed material transport in the Virgin River, Utah. Water Resour. Res. 2000, 36, 585–596.

[CrossRef]41. Komar. Flow competence of the hydraulic parameters of floods. In Floods: Hydrological, Sedimentological

and Geomorphological Implications; Beven, K., Carling, P., Eds.; John Wiley and Sons: Chichester, UK, 1989;pp. 107–134.

42. Buffington, J.M.; Montgomery, D.R. A systematic Analysis of Eight Decades of Incipient Motion Studies,with special Reference to Gravel-Bedded Rivers. Water Resour. Res. 1997, 33, 1993–2029. [CrossRef]

43. Wilcox, P.R.; Crowe, J.C. Surface-Based Transport Model for Mixed-Size Sediment. J. Hydraul. Eng. 2003, 129,120–128.

44. Thuc, T. Two-Dimensional Morphological Computations Near Hydraulic Structures. Ph.D. Thesis, AsianInstitute of Technology, Bangkok, Thailand, 1991.

45. Rahuel, J.L.; Holly, F.M.; Chollet, J.P.; Belleudy, P.J.; Yang, G. Modeling of riverbed evolution for bedloadsediment mixtures. J. Hydraul. Eng. 1989, 115, 1521–1542. [CrossRef]

46. Gaeuman, D.; Sklar, L.; Lai, Y.G. Flume experiments to constrain bedload adaptation length. J. Hydrol. Eng.2014, 20, 06014007. [CrossRef]

47. Philips, B.C.; Sutherland, A.J. Spatial lag effect in bed load sediment transport. J. Hydraul. Res. 1989, 27,115–133. [CrossRef]

48. Han, Q.W. A study on the non-equilibrium transportation of suspended load. In Proceedings of the 1stInternational Symposium on River Sedimentation, IRTCES, Beijing, China, 24–29 March 1980.

49. Han, Q.W.; He, M. A mathematical model for reservoir sedimentation and fluvial processes. Int. J.Sediment Res. 1990, 5, 43–84.

Water 2020, 12, 1032 21 of 21

50. Lai, Y.G.; Weber, L.J.; Patel, V.C. Non-Hydrostatic Three-Dimensional Method for Hydraulic FlowSimulation—Part I: Formulation and Verification. J. Hydraul. Eng. ASCE 2003, 129, 196–205. [CrossRef]

51. Yanenko, N.N. The Method of Fractional Steps; Springer: New York, NY, USA, 1971.52. Soni, J.P. Laboratory study of aggradation in alluvial channels. J. Hydrol. 1981, 49, 87–106. [CrossRef]53. Ashida, K.; Michiue, M. An investigation of river bed degradation downstream of a dam. In Proceedings of

the 14th IAHR Congress, Paris, France, 29 August–3 September 1971; Volume 3, pp. 247–256.54. Struiksma, N.; Olsen, K.W.; Flokstra, C.; De Vriend, H.J. Bed deformation in curved alluvial channels.

J. Hydraul. Res. 1985, 23, 57–79. [CrossRef]55. Tetra Tech, Inc. Sediment Transport Analysis Total Load Report, June 2008, SO-1470.5 and EB-10 Lines, Tetra Tech

Project No. T22541; Albuquerque Area Office, U.S. Bureau of Reclamation: Albuquerque, NM, USA, 2008.56. Lai, Y.G.; Collins, K.L. Sediment Plug Formation on the Rio Grande: A Review of Numerical Modeling; Technical

Report No. SRH-2008-06; Technical Service Center, Bureau of Reclamation: Denver, CO, USA, 2008.57. Boroughs, C.B. Sediment Plug Computer Modeling Study, Tiffany Junction Reach, Middle Rio Grande Project;

Albuquerque Aera Office, U.S. Bureau of Reclamation: Albuquerque, NM, USA, 2005; 122p.58. Collins, K.L. 2006 Middle Rio Grande Maintenance Modeling: San Antonio to Elephant Butte Dam. U.S. Bureau of

Reclamation; Technical Service Center, Sedimentation and River Hydraulics Group: Denver, CO, USA, 2006.59. Bauer, T.R. 2006 Bed Material Sampling on the Middle Rio Grande, NM, U.S. Bureau of Reclamation; Technical

Service Center, Sedimentation and River Hydraulics Group: Denver, CO, USA, 2006.

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).